Learning Objectives
Solve a linear equation using equivalent steps.
Recognize equations with one, no, or infinitely many solutions.
Solve and interpret a linear inequality, including a negative coefficient.
Prerequisites: signed arithmetic, fractions, and the distributive property. These original practice questions develop algebra skills for ACT preparation; they are not official exam questions or a complete coverage statement.
1. Preserve Equality at Every Step
An equation asserts that two expressions have the same value. To find the values of the variable that make this true, use operations that preserve the solution set: add or subtract the same quantity on both sides, or multiply or divide both sides by the same nonzero constant.
Worked Example: Variables on Both Sides
Solve 3(2x − 1) + 4 = 2x + 17. Distribute and combine: 6x + 1 = 2x + 17.
Subtract 2x and then 1 from both sides: 4x = 16. Divide by 4 to get x = 4.
Check in the original equation: 3(8 − 1) + 4 = 25 and 2(4) + 17 = 25. Both sides agree.
2. Simplification Can Remove the Variable
Simplified result | Interpretation |
|---|---|
4x = 16 | One solution: x = 4 |
0 = 5 | No solution: a contradiction |
0 = 0 | Every real x works: an identity |
When comparing ax + b = cx + d, subtract cx and b to obtain (a − c)x = d − b. If a ≠ c, there is one solution. If a = c, compare b and d: equal constants give an identity, and unequal constants give no solution.
Example: A Parameter
For what value of k does kx + 2 = 5x + 8 have no solution?
Make the x coefficients equal: k = 5. Then the equation becomes 5x + 2 = 5x + 8, or 2 = 8, which is impossible.
If k ≠ 5, the variable coefficient does not cancel and the equation has one solution.
3. Inequalities Describe a Set of Values
Solving an inequality uses many of the same steps as solving an equation. The extra rule is that multiplying or dividing both sides by a negative number reverses the inequality direction. Addition and subtraction do not reverse it.
Worked Example: A Negative Divisor
Solve 7 − 3x ≥ 19. Subtract 7 to obtain −3x ≥ 12.
Divide both sides by −3 and reverse the inequality: x ≤ −4.
Check the boundary: x = −4 makes both sides equal to 19. Check an included value: x = −5 gives 22 ≥ 19. An excluded value x = 0 gives 7 ≥ 19, which is false.
For a compound inequality such as 2 < 3x − 1 ≤ 11, apply each operation to all three parts. Add 1 to obtain 3 < 3x ≤ 12, then divide by 3: 1 < x ≤ 4. The lower endpoint is excluded and the upper endpoint is included.
4. Match the Requested Quantity
Sometimes a question asks for an expression rather than the variable. If 2x + 3 = 11, then 6x + 9 is three times the entire left side, so it equals 33. You can also solve for x and substitute, but identifying the expression relationship can shorten the calculation.
Common Mistakes
Distribute a negative sign to every term inside parentheses.
An equation that simplifies to 0 = 0 does not mean x = 0; it means all real x satisfy it.
When dividing an inequality by a negative number, reverse the relation symbol, not the signs of unrelated terms.
5. Practice
1. Solve (x + 5)/3 = 2x − 5.
2. Classify the solutions of 4(x − 2) = 4x + 1.
3. Solve −2x + 5 < 13.
4. If 3x − 2 = 7, find 9x − 6.
Worked Solutions
1. Multiply every term by 3: x + 5 = 6x − 15. Then 20 = 5x and x = 4. Both original sides equal 3.
2. Expansion gives 4x − 8 = 4x + 1, so −8 = 1. There is no solution.
3. −2x < 8, so division by −2 gives x > −4. The endpoint −4 is excluded because the inequality is strict.
4. Multiply the given equation by 3. The requested expression equals 21.
6. Summary
- Preserve the solution set by applying valid operations to both sides.
- Classify a simplified equation before assuming there is a single answer.
- Reverse inequalities only when a negative multiplication or division requires it.
- Check the expression or set of values actually requested.